The defective conditional in mathematics

Vidal, Mathieu

doi:10.1080/11663081.2014.911540

Cited by 5 publications

(2 citation statements)

References 18 publications

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“…3 Moreover, properties such as contraposition and transitivity, which are invalid for the variably strict conditional but valid for the strict conditional, seem to be essential to many inferences routinely used in reasoning. After three decades in which the Stalnaker-Lewis theory was viewed as the standard analysis of conditionals, the strict conditional started to gain attention again [12,13,18,19,27,32,33,74,77,78,81]. 4 The third view of conditionals discussed by Sextus Empiricus, which has been attributed to Chrysippus, states that a conditional is true when the contradictory of the consequent is incompatible with the antecedent.…”

Section: Introductionmentioning

confidence: 99%

The Implicative Conditional

Raidl,

Gomes

2023

J Philos Logic

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This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q . We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut.

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Section: Introductionmentioning

confidence: 99%

The Implicative Conditional

Raidl,

Gomes

2023

J Philos Logic

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“…The main advantage of the above valuation is to afford a stronger characterization of it; stronger, in the sense that some of the theorems where the material implication of classical logic essentially occurs do not hold in AR 4 . Our logical constant is a case of "defective" conditional [14], which means that the logical relation cannot be satisfied unless the antecedent is said true or, equivalently, accepted by the speaker. Such a feature cancels any form of the highly counterintuitive paradoxes of material implication, by virtue of which the whole relation holds once the antecedent is false.…”

Section: Strong Conditionalmentioning

confidence: 99%

The Football of Logic

Schang

2017

Studia Humana

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An analogy is made between two rather different domains, namely: logic, and football (or soccer). Starting from a comparative table between the two activities, an alternative explanation of logic is given in terms of players, ball, goal, and the like. Our main thesis is that, just as the task of logic is preserving truth from premises to the conclusion, footballers strive to keep the ball as far as possible until the opposite goal. Assuming this analogy may help think about logic in the same way as in dialogical logic, but it should also present truth-values in an alternative sense of speech-acts occurring in a dialogue. The relativity of truth-values is focused by this way, thereby leading to an additional way of logical pluralism.

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Generalized Correspondence Analysis for Three-Valued Logics

Petrukhin

2018

Log. Univers.

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The defective conditional in mathematics

Cited by 5 publications

References 18 publications

The Implicative Conditional

The Implicative Conditional

The Football of Logic

Generalized Correspondence Analysis for Three-Valued Logics

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